As the article explains, the force on a magnetic dipole in a non-uniform magnetic field is

$$\vec F=\vec\nabla(\vec m \cdot \vec B)= (\vec m\cdot\vec\nabla)\vec B.$$

The final expression involves the directional derivative of the magnetic field in the direction of the dipole moment. This is what it means by “proportional to the magnetic field gradient”. If you want, you can think of this as $\vec m\cdot(\vec\nabla\vec B)$ where the parenthesized expression is a tensor which is the gradient of the magnetic field.

As the article explains, the force on a magnetic dipole in a non-uniform magnetic field is

$$\vec F=\vec\nabla(\vec m \cdot \vec B)= (\vec m\cdot\vec\nabla)\vec B.$$

The final expression involves the directional derivative of the magnetic field in the direction of the dipole moment. This is what it means by “proportional to the magnetic field gradient”. 

If you want, you can write this as $\vec m\cdot(\vec\nabla\vec B)$ where the parenthesized expression is a tensor which is indeed the gradient of the magnetic field. The gradient of a scalar field is a vector field; the gradient of a vector field is a (rank-2) tensor field.

The force on a magnetic dipole in a non-uniform magnetic field is

$$\vec F=\vec\nabla(\vec m \cdot \vec B)= (\vec m\cdot\vec\nabla)\vec B.$$

As the article explains, the force on a magnetic dipole in a non-uniform magnetic field is

$$\vec F=\vec\nabla(\vec m \cdot \vec B)= (\vec m\cdot\vec\nabla)\vec B.$$

The final expression involves the directional derivative of the magnetic field in the direction of the dipole moment. This is what it means by “proportional to the magnetic field gradient”. If you want, you can think of this as $\vec m\cdot(\vec\nabla\vec B)$ where the parenthesized expression is a tensor which is the gradient of the magnetic field.